1. Field of the Invention
The present invention is directed to methods for loosely coupling a stress analysis system (geomechanical simulator) and a conventional reservoir simulator. In particular, the methods of the invention provide the adjusting of the flow equation of the conventional reservoir simulator, considering the variation of the stress state of the rock. Therefore, the solution obtained using the methods according to the invention in a loose, iterative coupling system, when convergence is reached the results obtained by this solution are close to those of the full coupling system. The present invention is also directed to a system for implementing the invention provided on a digitally readable medium.
2. Prior Art
Recently, there has been great the interest in the study of loose coupling between stress analysis systems and conventional reservoir simulators. The problem of flow and stress can be coupled by using different coupling systems, the three main systems are: full coupling, loose iterative coupling and loose explicit coupling.
In full coupling, the set of equations governing the hydro-mechanical problem is solved simultaneously by a single simulator, featuring the more rigorous formulation of coupling.
In the loose iterative coupling (or two-way coupling), the flow and stress equations are solved separately and sequentially for each time interval. FIG. 1(a) illustrates this system, the information are exchanged in the same time interval, between the conventional reservoir simulator and the stress analysis system until it reaches the convergence of an unknown variable, for example, pressure.
In loose explicit coupling (or one-way coupling), only the conventional reservoir simulator sends information (variation of pore pressure) to the stress analysis system. FIG. 1(b) illustrates this system, no information is sent from the stress analysis system to the conventional reservoir simulator, so the flow problem is not affected by changing in stress state in the reservoir and surrounding rocks.
The four main studies published in the literature concerning loose coupling between stress analysis systems and conventional reservoir simulators are briefly described below.
Settari and Mounts1 presented one of the first studies about loose coupling of a conventional reservoir simulator (DRS-STEAM) and a stress analysis system (FEM3D). The authors presented an algorithm for loose iterative coupling and the porosity, depending on the pore pressure field and the stress state, which was the only coupling parameter used.
Mainguy and Longuemare4 had presented three equations to correct the equation for porosity frequently used in conventional reservoir simulator to take into account the variation in stress state. Again, the porosity was the only coupling parameter used in the loose coupling.
Dean et al.3 showed results of three hydro-mechanical coupling system: loose explicit, loose iterative and full. In the coupling system called by the authors as loose explicit, the porosity is evaluated in two ways: 1) considering the variation of the pore pressure field and stress state; 2) considering the variation of pore pressure field and the calculation of a new compressibility, evaluated by the simple loading hypothesis, such as uniaxial deformation. Only the porosity was considered as coupling parameter in both loose coupling systems. The three coupling systems were implemented in the program Acres (ARCOS's Comprehensive Reservoir Simulator).
Samier and Gennaro2 proposed a new loose iterative coupling system. In this new coupling system, the iterations are not performed in time intervals, they are performed over the complete time of analysis. The effect of stress analysis is introduced in the porosity through a functionality of the conventional reservoir simulator called pore volume multiplier. Again, the porosity was the only coupling parameter considered.
From reading these four studies, it can be concluded that only the porosity is used as coupling parameter between the stress analysis system and the conventional reservoir simulator.
The Governing Equations
In the present invention, the governing equations are formulated using continuum mechanics, which analyzes the mechanical behavior of modeled materials as a continuous (solid, liquid and gases).
The Governing Equations for the Flow Problem
The flow equation is obtained from the mass conservation law. The mass conservation law can be represented mathematically by the equation:
                                          -            ∇                    ·                      (                                          ρ                f                            ⁢              v                        )                          =                                            ∂                                                                                  ∂              t                                ⁢                      (                                          ρ                f                            ⁢              ϕ                        )                                              (        1        )            wherein v is the velocity vector (L/t), ρf is the specific mass of the fluid (m/L3) and φ is the porosity.
Darcy's law states that the rate of percolation is directly proportional to the pressure gradient:
                    v        =                              -                          k              μ                                ⁢                      ∇            p                                              (        2        )            wherein k is the absolute permeability (L2), μ is the viscosity (m/Lt) and p the pore pressure (m/Lt2).
In geomechanical terms, the difference in the development of the flow equation used in conventional reservoir simulation and in the full coupling system is in development of the formulation adopted to evaluate the porosity. In conventional reservoir simulation, the variation of porosity can be related to the variation of pore pressure through the compressibility of rock, using a linear relation.φ=φ0[1+cr(p−p0)]  (3)wherein φ0 is the inicial porosity, p0 is the initial pore pressure and cr is the compressibility of the rock that can be calculated as:
                              c          r                =                                            1                              ϕ                0                                      ⁢                                          ∂                ϕ                                            ∂                p                                              =                                    1                              V                p                0                                      ⁢                                          ∂                                  V                  p                                                            ∂                p                                                                        (        4        )            wherein Vp0 is the porous volume in the initial configuration and Vp is the porous volume in the final configuration.
The fluid compressibility is taken into account in conventional reservoir simulation using the equation:ρf=ρf0[1+cf(p−p0)]  (5)wherein ρ0f is the initial specific mass of the fluid and cf is the fluid compressibility which relates the variation of specific mass with the variation of pore pressure.
                              c          f                =                                            1                              ρ                f                0                                      ⁢                                          ∂                                  ρ                  f                                                            ∂                p                                              =                      1                          K              f                                                          (        6        )            wherein Kf is the modulus of volumetric deformation of the fluid.
Introducing equation (3) and (5) in equation (1), the variation of porosity and specific mass of the fluid over time (accumulation term) can be considered in the simulation of conventional reservoirs. The final form of the flow equation can be written as:
                                                        (                                                                    c                    f                                    ⁢                                      ϕ                    0                                                  +                                                      c                    r                                    ⁢                                      ϕ                    0                                                              )                        ⁢                                          ∂                p                                            ∂                t                                              -                                    k              μ                        ⁢                                          ∇                2                            ⁢              p                                      =        0                            (        7        )            
To an elastic, linear and isotropic analysis, the expression of the variation of porosity used in the full coupling system of is comprised of four components that contribute to fluid accumulation (Zienkiewicz et al.5).
a) The variation of volumetric deformation: −dεv;
b) The variation due to compression of the solid matrix by pore pressure: (1−n) dp/KS;
c) The variation due to compression of the solid matrix by effective stress: −KD/KS (dεv+dp/Ks);
d) The variation due to compression of the fluid by pore pressure: ndp/Kf.
The equation of the variation of porosity is obtained from the sum of the above four components:
                    ϕ        =                              ϕ            0                    +                      α            ⁡                          (                                                ɛ                  v                                -                                  ɛ                  v                  0                                            )                                +                                    1              Q                        ⁢                          (                              p                -                                  p                  0                                            )                                                          (        8        )            wherein ε0v is the initial volumetric deformation (solids+pores) and εv is the final volumetric deformation (solids+pores). The parameter Q of Biot6 is written as:
                              1          Q                =                                                            ϕ                0                                            K                f                                      +                                          α                -                                  ϕ                  0                                                            K                S                                              =                                                    c                f                            ⁢                              ϕ                0                                      +                                          c                S                            ⁡                              (                                  α                  -                                      ϕ                    0                                                  )                                                                        (        9        )            wherein cs is the compressibility of solid matrix (cs=1/Ks) ans Ks is the modulum of volumetric deformation of solid matrix.
The parameter α of Biot6 is written in terms of the module of volumetric deformation of the rock and the pores KD and the volumetric deformation modulus of solid matrix KS (Zienkiewics et al5).
                    α        =                  1          -                                    K              D                                      K              S                                                          (        10        )            wherein KD is the volumetric deformation modulus associated to the drained constitutive matrix tangent C.
                              K          D                =                                                            m                T                            ⁢              Cm                        9                    =                      E                          3              ⁢                              (                                  1                  -                                      2                    ⁢                    v                                                  )                                                                        (        11        )            wherein m is the identity matrix, E is the Young modulus and v is the Poisson's ratio.
Insert equations (5) and (8) in equation (1), the variation of porosity that takes into account the variation of stress state is considered in terms of accumulation, resulting in the flow equation of full coupling system.
                                                        [                                                                    c                    f                                    ⁢                                      ϕ                    0                                                  +                                                      c                    S                                    ⁡                                      (                                          α                      -                                              ϕ                        0                                                              )                                                              ]                        ⁢                                          ∂                p                                            ∂                t                                              -                                    k              μ                        ⁢                                          ∇                2                            ⁢              p                                      =                              -            α                    ⁢                                    ∂                              ɛ                v                                                    ∂              t                                                          (        12        )            The Governing Equations for the Problem Geomechanical
The balance equation is written as:∇·σ+ρb=0  (13)wherein δ is the total stress tensor and ρ is the total density of the composition, that is:ρ=φρf+(1−φ)ρS  (14)wherein ρs is the density of solid matrix.
The tensor of deformations ε can be written in terms of the vector displacement u as:ε=½[∇u+(∇u)T]  (15)
The principle of Terzaghi effective stress is written as:σ′=σ−αmp  (16)wherein δ′ is the effective stress tensor.
The effective stress tensor is related to the tensor of deformations through the drained constitutive tangent c matrix.σ′=C:ε  (17)
Introducing equations (15), (16) and (17) in equation (13), the balance equation coupled in terms of displacement and pore pressure can be written as:
                                          G            ⁢                                          ∇                2                            ⁢              u                                +                                    G                              1                -                                  2                  ⁢                  v                                                      ⁢                                          ∇                ∇                            ·              u                                      =                  α          ⁢                      ∇            p                                              (        18        )            wherein G is the shear module.The Governing Equations for the Partial Coupling Scheme
FIG. 2 shows the assembly of the governing equations of the loose coupling system, the flow equation (7) is obtained from the conventional reservoirs simulation and the mechanical behavior is governed by the balance equation (12) written in terms of displacement and pore pressure, the same used in the full coupling system. The challenge of this coupled problem is to get from flow equation (7) of conventional reservoir simulation the same response of flow equation (12) of the full coupling system. In general, conventional reservoir simulators are closed-source software (proprietary software), hampering the process of loose coupling, so it is necessary to use external artifices to reshape the flow equation (7) conventional reservoir simulator.
Comparing the flow equations (7) and (12) can be observed that the terms cfφ0∂p/∂t and k/μ∇2p are common in the equations. The term crφ0∂p/∂t is found only in the equation (7) and the terms cs(α−φ0)∂p/∂t and α∂εv/∂t are found only in equation (12).
The literature references cited above do not anticipate or even suggest the invention scope. They are listed below in more detail for simple verification.    (1) A. Settari and F. M. Mourits, “Coupling of Geomechanics and Reservoir Simulation Models”, Computer Methods and Advances in Geomechanics, Siriwardane & Zanan (Eds), Balkema, Rotterdam (1994).    (2) P. Samiei and S. De Gennaro, “Practical Interative Coupling of Geomechanics with Reservoir Simulation”, SPE paper 106188 (2007).    (3) R. H. Dean, X. Gai, C. M. Stone, and S. Mikoff, “A Comparison of Techniques for Coupling Porous Flow and Geomechanics”, paper SPE 79709 (2006).    (4) M. Mainguy and P. Longuemare, “Coupling Fluid Flow and Rock Mechanics: Formulations of the Partial Coupling Between Geomechanical and Reservoir Simulators”, Oil & Gas Science and Technology, Vol 57, No. 4, 355-367 (2002).    (5) O. C. Zienkiewicz, A. H. C. Chan, M. Shepherd, B. A. Schrefler, and T. Shiomi, “Computational Geomechanics with Special Reference to Earthquake Engineering”, John Wiley and Sons, (1999).    (6) M. A. Biot, “General Theory of Three-Dimensional Consolidation”, J. Appl. Phys., Vol 12, 155-164, (1940).
Prior art patent has documents relating to this topic, in which the most significant are described below.
Document U.S. Pat. No. 7,386,431 describes a system and method for modeling and simulating the event of “fracture” in oil wells, especially the phenomena known as “interfacial slip” or “debonding” between adjacent layers of the Earth formation.
Document U.S. Pat. No. 7,177,764 describes a method to calculate stress around a fault while computing the prediction of the rock stress/fluid flow by means of the conservation of momentum. The method takes into account the presence of a multi-phase flow, where the number of flowing fluid phases is between 1 and 3.
Document WO 2008/070526 describes a method for calculation and simulating fluid flow in a fractured subterranean reservoir from the combination of discrete fracture networks and homogenization of small fractures.
The present invention differs from these documents by providing alternatives that allow approximating the flow equation of the conventional reservoir simulation to the flow equation of the full coupling system, removing the rock compressibility effect (crφ0∂p/∂t) and adding the volumetric deformation effect of rock and the pores (α∂εv/∂t).
Therefore, can be observed that none of the mentioned documents disclose or even suggest at the concepts of the present invention, so that it presents the requirements for patentability.